Accelerating Langevin Dynamics with Weighted Laplacian Flow: Global Well-Posedness and Exponential Convergence
Abstract
We study an accelerated Langevin equation in which the classical overdamped Langevin drift is supplemented by a nonlocal gradient velocity determined through a target-weighted Poisson equation. The target p is only assumed to be strictly positive and need not be normalized. Writing the dynamics in terms of ω=log(p/q) yields a coupled parabolic-elliptic system whose parabolic component contains the usual Langevin diffusion, whereas the elliptic component generates a global transport. The system is invariant under multiplication of p by a positive constant. On a bounded smooth domain with homogeneous Neumann boundary conditions, we establish global existence and uniqueness of classical solutions for strictly positive C2+α target weights and compatible C2+α initial data. For long-time behavior, the oscillation of ω decays at the explicit rate e−t and the solution density exponentially converges to target density. Moreover, an exact entropy-dissipation identity holds relative to the target density, in which the relative Fisher information and the Jeffreys divergence give the two dissipation mechanisms associated with the Langevin and weighted-Poisson components. In particular, KL divergence also converges to zero with rate e−t without a convexity assumption on the target.
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BibTeX
@misc{cai2026accelerating,
title = {{Accelerating Langevin Dynamics with Weighted Laplacian Flow: Global Well-Posedness and Exponential Convergence}},
author = {Ziyue Cai and Weiye Gan and Borui Miao and Zuoqiang Shi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35892},
eprint = {2610.35892},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35892},
}
RIS
TY - UNPB
TI - Accelerating Langevin Dynamics with Weighted Laplacian Flow: Global Well-Posedness and Exponential Convergence
AU - Ziyue Cai
AU - Weiye Gan
AU - Borui Miao
AU - Zuoqiang Shi
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.35892
UR - https://arxiv.org/abs/2610.35892
AB - We study an accelerated Langevin equation in which the classical overdamped Langevin drift is supplemented by a nonlocal gradient velocity determined through a target-weighted Poisson equation. The target p is only assumed to be strictly positive and need not be normalized. Writing the dynamics in terms of ω=log(p/q) yields a coupled parabolic-elliptic system whose parabolic component contains the usual Langevin diffusion, whereas the elliptic component generates a global transport. The system is invariant under multiplication of p by a positive constant. On a bounded smooth domain with homogeneous Neumann boundary conditions, we establish global existence and uniqueness of classical solutions for strictly positive C^2+α target weights and compatible C^2+α initial data. For long-time behavior, the oscillation of ω decays at the explicit rate e^−t and the solution density exponentially converges to target density. Moreover, an exact entropy-dissipation identity holds relative to the target density, in which the relative Fisher information and the Jeffreys divergence give the two dissipation mechanisms associated with the Langevin and weighted-Poisson components. In particular, KL divergence also converges to zero with rate e^−t without a convexity assumption on the target.
ER -